Before the description each chapter, we emphasize that all theory which are borrowed is marked with a reference. If there are things that are inspired by external sources this is also mentioned.
• Chapter 2: Stochastic theory beyond an introductory course is pre-sented in this chapter. Originally my supervisor Fred Espen proposed to study an exponential negative subordinator stochastic volatility model, driven by a compound Poisson process. However, inspired by the derivation of LIBOR forward rates in [Fil09], where the derivation is based on an Itô process, I soon decided to derive an extended model based on an Itô-Lévy process (as [ØS07] calls them). This chapter is a result of that decision. It is an attempt to do a thorough but efficient introduction of Itô-Lévy processes. All material in this chapter is achieved from external sources, but the structure and discussions around all mathematical statements are worked out by me. As my knowledge of Lévy processes from before was almost equal to zero, this chapter has been developed over several months. It has been challenging to understand every aspect of the presented theory, and I had to use a lot of sources to be able to give an introduction as complete as I wanted.
• Chapter 3:As this thesis focuses on LIBOR forward rates, an intro-duction to the interest rate market was needed. I have tried to give an introduction which gives both a mathematical and financial/eco-nomical aspect of the topic. This was done to be sure that I was able to use the necessary mathematical approaches, and at the same time understand the application of it. All theory in this chapter is inspired by external sources.
• Chapter 4:The derivation of the extended LIBOR forward rate model in this chapter is highly inspired by the derivation in [Fil09]. That is, the approach is similar in the way that I define an instantaneous forward rate model, derives a zero-coupon bond price model from it, and then find an expression for its discounted model. From that model, and its characteristics, the LIBOR forward rate model and the LIBOR market model are derived. The difference between my derivation and the one done in [Fil09], is that the instantaneous forward rate model in [Fil09] is driven by an Itô process with deterministic coefficients, while in this thesis the instantaneous forward rate model is driven by
1.2. Chapter overview: approach and contributions an Itô-Lévy process. This adds a lot of extra work in more complicated theory and heavier expressions. Because the model is derived in this way, I have chosen to call it the LIBOR forward rate in the HJM-Lévy framework. In the end I achieve a flexible model, and the model derived in [Fil09] is a special case of it. All calculations and discussions in this chapter are performed by myself.
• Chapter 5: If the extended LIBOR forward rate model derived in Ch. 4 is considered with zero jump part, it is reduced to a geometric Brownian motion with stochastic volatility. As mentioned above, my supervisor initially proposed to analyze a specific stochastic volatility model in this thesis. Based on these facts I decided to analyze the extended LIBOR forward rate model further, with zero jump part, and the proposed stochastic volatility. The volatility model he proposed is on skeleton form an exponential negative subordinator process. I deduce characteristics of this stochastic volatility, and of the LIBOR forward rate driven by it. It was cool to realize (even if it is quite obvi-ous now) that all the characteristics are functions of the exponential moments and joint exponential moments of the subordinator. One pos-sible choice for the subordinator is a non-Gaussian OU subordinator (nGOUS). The chapter is introduced with calculations of the expo-nential moments and joint expoexpo-nential moments of this subordinator, to set the stage. Except from the calculations of the characteristic function of the nGOUS, which is inspired by calculations done in a course I attended at UiO, all other calculations is done exclusively by me.
• Chapter 6:I wanted to get an impression of how a special case of the LIBOR forward rate in the HJM-Lévy framework could behave. It was natural to continue the analysis on the geometric Brownian motion model with stochastic volatility, which was considered in Ch. 5. To be able to analyze the model further, a specific Poisson random measure had to be chosen for the nGOUS. Earlier I have worked with compound Poisson processes (CPPs) with exponential jumps in a course which I attended at UiO. Because of this I chose to consider this specific Poisson random measure with exponential jumps. The derivation of the characteristic function of the CPP nGOUS with exponential jumps was is inspired by the lecture notes from the above mentioned course.
Other than that calculations are done independent of any source of inspiration.
• Chapter 7:The LIBOR market model was derived such that options could be prices on them in a simple way. That is, when the LIBOR forward rate is log-normally distributed and a martingale, Black’s formula (Prop. 3.3.1) can be used to find the fair price. That is not the case for the LIBOR forward rate in the HJM-Lévy framework, except for the special case when it is a geometric Brownian motion with deterministic stochastic volatility. Based on this I wanted to explore if it was possible to derive a general caplet valuation formula for the LIBOR forward rate in the HJM-Lévy framework. In this chapter a Fourier transformation method is used. The method is derived in
[EGP10], and the two first sections in this chapter are very much inspired by that publication.
• Chapter 8: I though that there had to be possible to use another approach than Fourier transformations to derive a caplet valuation formula for the special case of the LIBOR forward rate in the HJM-Lévy framework, that was considered in Ch. 5. That is, Black’s formula is based on the Black-Scholes approach for log-normally dis-tributed random variables. In my special case from Ch. 5, the LIBOR forward rates are not log-normally distributed, but at least they are driven by a Brownian motion with stochastic volatility. Inspired by the classical Black-Scholes derivation in [Ben04] I therefore decided to derive a similar formula for models with stochastic volatilities.
Further work in that chapter is all done by me.
Chapter 2
THEORY AND NOTATION: ITÔ-LÉVY PROCESSES
In this chapter we are going to introduce theory relevant for this thesis, as well as notation and assumptions used throughout. The presented theory is mainly obtained from and inspired by [ØS07] and [App04]. We assume that the reader is familiar with basic theory in mathematical and stochastic analysis. If nothing else mentioned we will always work with function values inR,t∈[0,T]andω ∈Ω. To set the framework, we define a com-plete filtered probability space(Ω,F,{Ft}t≤T, P)whereΩis an appropriate sample space,Fis aσ-algebra of subsets ofΩ,{Ft}t≤T is a filtration on the measurable space(Ω,F), andP is the market probability measure onF.
This probability space will be used throughout the thesis, with a variation in which probability measure we work with respect to.
Let{X(t)}t≤T be a stochastic process in the sense thatX(t, ω) : [0,T]× Ω→Ris aB ⊗ F-measurable function, whereBdenotes the Borelσ-algebra onR+. For simplicity, we introduce the following notation.
Notation 2.0.1.Stochastic processes{X(t)}t≤T are denoted byX(t). It will be clear from the context if we discuss the process or the measurable function.
Processes called Itô-Lévy processes are extensively used in this thesis.
We are going to introduce such processes through the theory of Lévy pro-cesses. The introduction will be fairly thorough, but efficient, such that the reader gets an impression of what conditions we have to use in forthcoming derivations and analyzes.
Definition 2.0.1 (Lévy processes, [App04]).LetX(t)be a stochastic process defined on the filtered probability space(Ω,F,{Ft}t≤T, P).X(t)is called a Lévy process if the following conditions are satisfied:
1. X(0) = 0a.s.;
2. X(t)has independent and stationary increments;
3. X(t)is continuous in probability, i.e. for all >0andt≥0
s→tlimP
X(s)−X(t) >
= 0.
Notation 2.0.2.Lévy processes are denoted byL(t).
There is a variety of different stochastic processes that are Lévy cesses. In the following section we will introduce Brownian motion pro-cesses. They are important examples of Lévy processes, and we will see
that they are key in representing general Lévy processes. The same applies to the Poisson random measure, which will be introduced later.
2.1 Brownian motion processes
Brownian motion processes are the most widely used Lévy processes. In this section they are introduced briefly, and we will consider some of their most important characteristics.
Definition 2.1.1 (Brownian motions, [App04]).A Lévy process L(t) is a Brownian motion if
1. L(t)is normally distributed for eacht≥0, with mean0and variance given by the process volatility and the applied time interval;
2. L(t)has continuous sample paths.
Notation 2.1.1.Brownian motions are denoted byW(t).
Integrals with respect to Brownian motions are called Itô integrals. For an Itô integral to exist, the integrand has to satisfy certain conditions.
Functions satisfying these conditions are contained in a function space which we will callV.
Definition 2.1.2 (The function spaceV, [Øks10]).LetV =V([0,T]×[0,T]) be the class of functionsf(t, T)such that
1. f(t, T, ω) : [0,T]×[0,T]×Ω→RisB2⊗ F-measurable;
2. (T, ω)→f(t, T, ω)isB([0,T])⊗ Ft-measurable for eacht≤ T; 3. Eh
RT
0 f(t, T)2dti
<∞.
We writeV([0,T]×[0,T]) :=V([0,T]2)for simplicity. If the function is not given by any parameterT, then the space is reduced toV([0,T]).
Geometric Brownian motions are SDE’s on the form dX(t)
X(t) =α(t, T)dt+σ(t, T)dW(t),
whereα(t, T)is integrable andσ(t, T)∈ V[(0,T)2]. Its solution is given by (see App. A.2)
X(t) =X(0) exp Z t
0
σ(s, T)dW(s) + Z t
0
α(s, T)−1
2σ2(s, T)
ds
! .
In no-arbitrage frameworks we work with processes on the form X(t) =X(0) exp
Z t 0
σ(s, T)dW(s)−1 2
Z t 0
σ2(s, T)ds
!
, (2.1)
2.1. Brownian motion processes because, under appropriate conditions, they are martingales with respect to the measure under whichW(t)is a Brownian motion. We will introduce two notations which are useful when we work with such processes. The first notation is a simplification of the semimartingale dynamics inside the exponential function in Eq. (2.1).
Notation 2.1.2 (Stochastic integral I, [Fil09]).We define the dynamics (f ◦W) :=f(t, T)dW(t)−1
2f2(t, T)dt,
wheref(t, T)∈ V([0,T]2)andW(t)is a Brownian motion process.
Next we introduce a notation which we call the stochastic exponential, also called the Doléans-Dade exponential, which represents the solution of a geometric Brownian motion. In Sect. 2.3 we will see that the stochastic exponential represents the solution of a geometric Itô-Lévy process as well, and that the geometric Brownian motion is a special case of the geometric Itô-Lévy process.
Notation 2.1.3 (Stochastic exponential, [Fil09]).LetdX(t)be a real valued stochastic process dynamics. Then the stochastic exponential is defined as
Et(X) = exp Z t
0
dX(s)
! .
Finally, we will present two important theorems called Itô isometry and Novikov’s condition. Itô isometry is invaluable when it comes to analyses of Brownian motion processes, as it states the connection between the expectation of squared Itô integrals and the expectation of classical time integrals.
Theorem 2.1.1 (Itô isometry, [Øks10]).Letf(u, T)∈ V([s, t]×[0,T]). Then
E
Z t
s
f(u, T)dW(u)
!2
=E
"
Z t s
f2(u, T)du
# ,
wheres≤t≤ T.
Novikov’s condition is important because it states a sufficient condition for the stochastic exponential, with respect to the dynamics given in Nota.
2.1.2, to be a martingale. In no-arbitrage frameworks we are dependent on the fact that the model representing the underlying discounted financial asset price actually is a martingale.
Theorem 2.1.2 (Novikov’s condition, [Øks10]).A sufficient condition for Et(λ◦W)
to be a martingale fort≤ T is that
E
exp 1